Monodromy of Kodaira fibrations of genus 3

A Kodaira fibration is a non‐isotrivial fibration f:S→B$f: S\rightarrow B$ from a smooth algebraic surface S to a smooth algebraic curve B such that all fibers are smooth algebraic curves of genus g. Such fibrations arise as complete curves inside the moduli space Mg$\mathcal {M}_g$ of genus g algebraic curves. We investigate here the possible connected monodromy groups of a Kodaira fibration in the case g=3$g=3$ and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of M3$\mathcal {M}_3$ parametrizing curves whose Jacobians have extra endomorphisms.